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Sunday, March 04, 2007

Derivation of Logistic Growth versus Groping in the dark

Since many peak oil analysts like to use Logistic growth to model peak, for Hubbert Linearization, etc., I thought to give the standard derivation of the equation a run through. Of course, the logistic formulation comes about from studies of population dynamics, where the rate of birth and death follows strictly from the size of the population itself. This makes sense from the point of view of a multiplying population, but not necessarily from inanimate pools of oil. In any case, the derivation starts with two assumptions, the birth and death rates:

B = B0 - B1*PD = D0 + D1*P

We base the entire premise on a the negative sign on the second term in the birth rate -- in the event of limited resources such as food, the birth rate can only decrease with size of population (and the death rate correspondingly increases).

The next step involves writing the equation for population dynamics as a function of time.

dP/dt = (B-D)*P

This provides the underpinnings for exponential growth, however critically modulated by the individual birth and death functions. So if we expand the population growth rate, we get:

dP/dt = (B0-B1*P-D0-D1*P)*P = (B0-D0)*P - (B1+D1)*P2

which matches the classic Logistic equation formulation:

dP/dt = rP*(1-P/Pinfinity)

Where Pinfinity becomes the carrying capacity of the environment. So the leap of faith needed to apply this to oil depletion comes about from analogizing population to a carefully chosen resource variable. The one that history has decided to select, cumulatively extracted oil, leads to the classical bell-shaped curve for instantaneous extraction rate, i.e. the derivitive dP/dt. (Note that we can throw out the death term because it doesn't really mean anything.)

I have always had issues with both the upward part of the logistic curve derivative and the decline part. Trying to rationalize why instantaneous production would initially rise proportionally to the cumulative production only makes sense if oil itself drove the exponential growth. But we know that oil does not mate with itself as biological entities would, so the growth really has to do with human population increase (or oil corporation growth) causing the exponential rise. That remains a big presumption to the model. The decline too has a significant interpretion hurdle as well. Why exactly the rate of growth after we start approaching and bypassing peak has that peculiar non-linear modifier doesn't make a lot of sense; the human population hasn't stabilized as of yet (even though oil company growth certainly has, technically declining significantly through mergers and acquisitions). We really have to face that a lot of apples and oranges assumptions flow into this interpretion.

In the end, using the Logistic curve only makes sense as a cheap heuristic, something that we can get a convenient analytical solution from. It fits into the basic class of solutions similar to the "drunk looking for his car-keys under the lamp-post" problem. Somebody asks the drunk why he chose to look under the lamp-post. "Of course, that's where the light is". I have fundamental problems with this philosophy and have made it a challenge to myself to seek something better; if that means groping around in the dark, what the heck.

Climate Primates

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"My life's the disease that could always change
With comparative ease, just given the chance
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I wait for the worth, digging for just one chance
As prospects diminish, as nightmares swell
Some pray for Heaven while we live in Hell" - E&TB

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